Breakup capillary number

Figure 9.9 Dimensionless critical droplet size for breakup (capillary number Ca<- = Crj a/r) as a function of viscosity ratio M of the dispersed to the continuous phase for two-dimensional flows in the four-roll mill. The data sets correspond from bottom to top to ff — 1.0(0), 0.8 (A), 0.6 (0), 0.4 (V), and 0.2 ( ), with a defined in Eq. (9-17). The fluids are those described in Fig. 9-7. The solid lines are the predictions of a small-deformation theory, while the dashed lines are for a large-deformation theory. The closed squares are from Rallison s (1981) numerical solutions (see also Rallison and Acrivos 1978). (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

To understand how the dispersed phase is deformed and how morphology is developed in a two-phase system, it is necessary to refer to studies performed specifically on the behavior of a dispersed phase in a liquid medium (the size of the dispersed phase, deformation rate, the viscosities of the matrix and dispersed phase, and their ratio). Many studies have been performed on both Newtonian and non-Newtonian droplet/medium systems [17-20]. These studies have shown that deformation and breakup of the droplet are functions of the viscosity ratio between the dispersity phase and the liquid medium, and the capillary number, which is defined as the ratio of the viscous stress in the fluid, tending to deform the droplet, to the interfacial stress between the phases, tending to prevent deformation ... 

The tensor L defines the character of the flow. The capillary number for the drop deformation and breakup problem is... 

Many authors have worked on drop deformation and breakup, beginning with Taylor. In 1934, he published an experimental work [138] in which a unique drop was submitted to a quasi-static deformation. Taylor provided the first experimental evidence that a drop submitted to a quasi-static flow deforms and bursts under well-defined conditions. The drop bursts if the capillary number Ca, defined as the ratio of the shear stress a over the half Laplace pressure (excess of pressure in a drop of radius R. Pl = where yint is the interfacial tension) ... 

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

Fig. 7.24 Breakup of a droplet of 1 mm diameter in simple shear flow of Newtonian fluids with viscosity ratio of 0.14, just above the critical capillary number. [Reprinted by permission from H. E.H. Meijer and J. M. H. Janssen, Mixing of Immiscible Fluids, in Mixing and Compounding of Polymers, I. Manas-Zloczower and Z. Tadmor, Eds., Hanser, Munich (1994).]...

Favelukis et al. (37,38) dealt with the problem of droplet deformation in exten-sional flow with both Newtonian and non-Newtonian Power Law model fluids, as wellas bubble breakup. For the Newtonian case, they find that as an inviscid droplet (or bubble) deforms, the dimensionless surface area is proportional to the capillary number... 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number... 

Bentley and Leal have measured droplet shapes and critical conditions for droplet breakup over a wide range of capillary numbers, viscosity ratios, and flow types. The flow type is conveniently controlled in an apparatus called a four-roll mill, in which a velocity field is generated by the rotation of four rollers in a container of liquid (see Fig. 1-15). By varying the rotation rate of one pair of rollers relative to that of a second pair, velocity fields ranging from planar extension to nearly simple shear can be produced near the stagnation point. 

The results plotted in Fig. 9-9 are for steady flow, or flow in which the strain rate is increased very slowly. If, instead, the flow rate is increased in a sudden step, then the capillary number at which droplet breakup occurs is reduced (Bentley and Leal 1986). If the flow rate is abruptly increased so that Ca > 2Cac, the droplets may deform into long cylinders before breaking (Elemans et al, 1993 Vinckier et al. 1997). 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

Note that in shear for A, = 1, the critical capillary number = 1, whereas for A, > 1, increases with X and becomes infinite for X > 3.8. This means that the breakup of the dispersed phase in pure shear flow becomes impossible for X > 3.8. This limitation does not exist in extensional flows. 

The mechanisms governing deformation and breakup of drops in Newtonian liquid systems are well understood. The viscosity ratio, X, critical capillary number, and the reduced time, t, are the controlling parameters. Within the entire range of X, it was found that elongational flow is more efficient than shear flow for breaking the drops. 

To make things more interesting, the experimental observations of De Bruijn [1989] seem to have contradicted the latter conclusion. The author found that the critical capillary number for viscoelastic droplets is always higher (sometimes much higher) than for Newtonian ones, whatever the -value De Bruijn concluded that drop elasticity always hinders drop breakup. 

Some authors report the next guide principles that may be applied for blend morphology after processing, (i) Drops with viscosity ratios higher than 3.5 cannot be dispersed in shear but can be in extension flow instead, (ii) The larger the interfacial tension coefficient, the less the droplets will deform, (iii) The time necessary to break up a droplet (Tj,) and the critical capillary number (Ca ) are two important parameters describing the breakup process, (iv) The effect of coalescence must be considered even for relatively low concentrations of the dispersed phase. 

Following the early work by Thorsen et al., focused on the formation of monodisperse aqueous droplets in an organic carrier fluid performed on a microfluidic chip, and then followed by others works, the breakup mechanism responsible of droplet formation was later analyzed by Garstecki et al. ° showing that when is order of 1 the dominant contribution to the dynamics of breakup at low capillary numbers is not dominated by shear stresses, but it is driven by the pressure drop across the emerging droplet. 

In T-junctions, the strong effects of conflnement exerted by the presence of walls in the microchannels, coupled with the importance of the evolution of the pressure field during the process of formation of a droplet, confers a quasistatic character of the collapse of the dispersed streams, while the separation of time scales between the slow evolution of the interface during breakup and fast equilibration of the shape of the interface itself via capillary waves, and of the pressure field in the fluids via acoustic waves, are the basis of the observed monodispersity of the droplets and bubbles formed in microfluidic systems at low values of the capillary number. 

In the squeezing mode, the breakup obeys the quasistatic model, with only a slight dependence of the diameter of the droplets on the value of the capillary number Ca." ... 

Figure 35.4 shows the variation of ellipticity with respect to the Weber, Reynolds, and capillary numbers at various axial locations. As observed fi-om these figures, the droplets are big close to the injector and the Ohnesorge numbers are small. The Weber number is much larger than the critical Weber number ( 6) and the drops undergo breakup. The deformation predicted by the above correlation... 

Unlike in NEMD models, the microstructures emerging due to competition between the breakup and coalescence processes can be studied by using DPD modeling. For example, in Figure 26.23, the four principal mechanisms, the same as those responsible for droplets breakup [ 118,119], can be observed in DPD simulation of the R-T instability. As shown in [116,119], moderately extended drops for capillary number close to a critical value, which is a function of dynamic viscosity ratio... 

It is clear that this phenomenon is phase morphology-dependent. Only in those blends where the minor phase is dispersed into sufficiently fine droplets, this phase has the opportunity to exhibit fractirMiated crystallization. Hence, only at low blend compositions and/or good matching viscosities of both phases (where the capillary number C predicts droplet breakup being dominant above coalescence) the occurrence of coincident crystallization is possible. 

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